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From Segmented Images to Good Quality Meshes using Delaunay Refinement
"... Abstract. This paper surveys Delaunaybased meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the socalled Delaunay refinement technique allows to mesh surfaces and volumes ..."
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Abstract. This paper surveys Delaunaybased meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the socalled Delaunay refinement technique allows to mesh surfaces and volumes bounded by surfaces, with theoretical guarantees on the quality of the approximation, from a geometrical and a topological point of view. Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly nonuniform) sizing field. We show how this general paradigm can be adapted to produce anisotropic meshes, i.e. meshes elongated along prescribed directions. Lastly, we discuss extensions to higher dimensions, and especially to spacetime for producing timevarying 3D models. This is also of interest when input images are transformed into data points in some higher dimensional space as is common practice in machine learning. 1
Meshes preserving minimum feature size
, 2011
"... The minimum feature size of a planar straightline graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an nvertex input, we give a triangulation (meshing) alg ..."
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The minimum feature size of a planar straightline graph is the minimum distance between a vertex and a nonincident edge. When such a graph is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. For an nvertex input, we give a triangulation (meshing) algorithm that limits degradation to only a constant factor, as long as Steiner points are allowed on the sides of triangles. If such Steiner points are not allowed, our algorithm realizes O(lg n) degradation. This result 5 answers a 14yearold open problem by Bern, Dobkin, and Eppstein.
The Cutting Pattern Problem for Tetrahedral Mesh Generation
"... Summary. In this work we study the following cutting pattern problem. Given a triangulated surface (i.e. a twodimensional simplicial complex), assign each triangle with a triple of ±1, one integer per edge, such that the assignment is both complete (i.e. every triangle has integers of both signs) a ..."
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Summary. In this work we study the following cutting pattern problem. Given a triangulated surface (i.e. a twodimensional simplicial complex), assign each triangle with a triple of ±1, one integer per edge, such that the assignment is both complete (i.e. every triangle has integers of both signs) and consistent (i.e. every edge shared by two triangles has opposite signs in these triangles). We show that this problem is the major challenge in converting a volumetric mesh consisting of prisms into a mesh consisting of tetrahedra, where each prism is cut into three tetrahedra. In this paper we provide a complete solution to this problem for topological disks under various boundary conditions ranging from very restricted one to the most flexible one. For each type of boundary conditions, we provide efficient algorithms to compute valid assignments if there is any, or report the obstructions otherwise. For all the proposed algorithms, the convergence is validated and the complexity is analyzed. Key words: cutting pattern, graph labeling, tetrahedral mesh, prism